Problem 77 - A Complete Resource Book in Chemistry for JEE Main

Question

In which of the following gaseous reaction, the value of \(K_{p}\) is less than \(K_{c}\) ? (a) \(\mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCI}_{3}+\mathrm{Cl}_{2}\) (b) \(2 \mathrm{SO}_{2}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{SO}_{3}\) (c) \(2 \mathrm{HI} \rightleftharpoons \mathrm{H}_{2}+\mathrm{I}_{2}\) (d) \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\)

Step-by-Step Solution

Answer

Reactions (b) and (d) have \(K_{p} < K_{c}\).

Step 1: Write the general formula relating Kp and Kc

The relationship between the equilibrium constants \(K_{p}\) and \(K_{c}\) for a gaseous reaction is given by \[ K_{p} = K_{c}(RT)^{\Delta n} \]where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in moles of gas, calculated as moles of gaseous products minus moles of gaseous reactants.

Step 2: Calculate \(\Delta n\) for each reaction

(a) \(\mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCl}_{3}+\mathrm{Cl}_{2}\): \[ \Delta n = (1 + 1) - 1 = 1 \](b) \(2 \mathrm{SO}_{2}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{SO}_{3}\): \[ \Delta n = 2 - (2 + 1) = -1 \](c) \(2 \mathrm{HI} \rightleftharpoons \mathrm{H}_{2}+\mathrm{I}_{2}\): \[ \Delta n = (1 + 1) - 2 = 0 \](d) \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\): \[ \Delta n = 2 - (1 + 3) = -2 \]

Step 3: Determine if \(K_{p} < K_{c}\)

For \(K_{p}\) to be less than \(K_{c}\), the term \((RT)^{\Delta n} < 1\).This means \(\Delta n\) must be negative. From Step 2:- Reaction (a) \(\Delta n = 1\), \(K_{p} > K_{c}\).- Reaction (b) \(\Delta n = -1\), \(K_{p} < K_{c}\).- Reaction (c) \(\Delta n = 0\), \(K_{p} = K_{c}\).- Reaction (d) \(\Delta n = -2\), \(K_{p} < K_{c}\).

Step 4: Identify which choices have \(K_{p} < K_{c}\)

Based on the calculations, reactions (b) and (d) have \(K_{p} < K_{c}\) since their \(\Delta n\) values are negative.- Reaction (b) has \(\Delta n = -1\)- Reaction (d) has \(\Delta n = -2\)

Key Concepts

Equilibrium Constant

The equilibrium constant is a crucial concept in chemistry that helps us understand chemical reactions in a state of balance. For a general reaction \\[ aA + bB \rightleftharpoons cC + dD \] \(K_c\) is the equilibrium constant expressed in terms of concentrations of products and reactants. It's calculated as follows:

• \(K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\)

This formula means that at equilibrium, the ratio of the concentrations of the products raised to their coefficients over the concentrations of the reactants raised to their coefficients remains constant. The value of \(K_c\) tells us which side of the reaction is favored at equilibrium. A large \(K_c\) (greater than 1) indicates that products are favored, while a small \(K_c\) (less than 1) suggests that reactants are favored.

Kp and Kc relationship

In reactions involving gases, we often use the equilibrium constant in terms of partial pressures, denoted as \(K_p\). The relationship between \(K_p\) and \(K_c\) is important for understanding the dynamics of a reaction under different conditions:

• The relationship can be expressed as: \[ K_p = K_c (RT)^{\Delta n} \]

Here, \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in moles of gas between products and reactants. This equation allows chemists to convert between \(K_p\) and \(K_c\) depending on available data and desired analysis. If \(\Delta n\) is zero, \(K_p\) equals \(K_c\), simplifying calculations.

Change in Moles of Gas (Δn)

The term \(\Delta n\) is essential in linking \(K_p\) and \(K_c\). It measures the change in the number of moles of gas as a reaction proceeds from reactants to products:

• \(\Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants}\)

For example, in the synthesis of ammonia: \[ \mathrm{N}_2 + 3\mathrm{H}_2 \rightleftharpoons 2\mathrm{NH}_3\] Calculating \(\Delta n\), we find: \(\Delta n = 2 - (1 + 3) = -2\). This negative \(\Delta n\) indicates fewer moles of gas on the product side. When \(\Delta n\) is negative, \((RT)^{\Delta n} < 1\), leading to \(K_p < K_c\). This shows how \(\Delta n\) affects the equilibrium conditions.

Ideal Gas Constant (R)

The ideal gas constant \(R\) is a vital part of the equation that connects \(K_p\) and \(K_c\). It's a universal constant in the equation \(K_p = K_c (RT)^{\Delta n}\) and provides the necessary link between energy and temperature across different gas laws.

• The value of \(R\) is approximately 8.314 J/mol⋅K when using SI units.

This constant allows the conversion between different units, such as converting energy measurements from joules to moles per kelvin. Using \(R\) is crucial for maintaining consistency in calculations involving gases and reactions at equilibrium. It ensures that all chemical calculations align with physical constants, helping chemists accurately predict and analyze reaction behaviors under various conditions.